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Theorem ofcval 26709
Description: Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfval.1  |-  ( ph  ->  F  Fn  A )
ofcfval.2  |-  ( ph  ->  A  e.  V )
ofcfval.3  |-  ( ph  ->  C  e.  W )
ofcval.6  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  B )
Assertion
Ref Expression
ofcval  |-  ( (
ph  /\  X  e.  A )  ->  (
( F𝑓/𝑐 R C ) `  X
)  =  ( B R C ) )

Proof of Theorem ofcval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ofcfval.1 . . . . 5  |-  ( ph  ->  F  Fn  A )
2 ofcfval.2 . . . . 5  |-  ( ph  ->  A  e.  V )
3 ofcfval.3 . . . . 5  |-  ( ph  ->  C  e.  W )
4 eqidd 2455 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
51, 2, 3, 4ofcfval 26708 . . . 4  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
65adantr 465 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
7 simpr 461 . . . . 5  |-  ( ( ( ph  /\  X  e.  A )  /\  x  =  X )  ->  x  =  X )
87fveq2d 5806 . . . 4  |-  ( ( ( ph  /\  X  e.  A )  /\  x  =  X )  ->  ( F `  x )  =  ( F `  X ) )
98oveq1d 6218 . . 3  |-  ( ( ( ph  /\  X  e.  A )  /\  x  =  X )  ->  (
( F `  x
) R C )  =  ( ( F `
 X ) R C ) )
10 simpr 461 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  X  e.  A )
11 ovex 6228 . . . 4  |-  ( ( F `  X ) R C )  e. 
_V
1211a1i 11 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
) R C )  e.  _V )
136, 9, 10, 12fvmptd 5891 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( F𝑓/𝑐 R C ) `  X
)  =  ( ( F `  X ) R C ) )
14 ofcval.6 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  B )
1514oveq1d 6218 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( F `  X
) R C )  =  ( B R C ) )
1613, 15eqtrd 2495 1  |-  ( (
ph  /\  X  e.  A )  ->  (
( F𝑓/𝑐 R C ) `  X
)  =  ( B R C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4461    Fn wfn 5524   ` cfv 5529  (class class class)co 6203  ∘𝑓/𝑐cofc 26705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-ofc 26706
This theorem is referenced by:  probfinmeasb  26979
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